Power Set – Explanation and Solved Examples

# Power Set – Explanation and Solved Examples

## Power Set Definition with Examples

A power set is a set of all sets that are subsets of a given set. In other words, a power set is the collection of all possible subsets of a set.

For example, the power set of the set {1, 2, 3} is the following set:

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{ {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} }

This set contains six sets: {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}.

## Define Power Set with Example

A power set is a set of all subsets of a given set.

For example, the power set of the set {1, 2, 3} would be:

{1, 2, 3}
{1, 2}
{1, 3}
{2, 3}
{1}
{2}
{3}

This is because the set {1, 2, 3} has six subsets: {1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, and {2}.

## Subsets

A subset is a group of elements that are part of a larger set. The smaller set is a part of the larger set, and can be identified by a number or letter. For example, the set {1, 2, 3, 4} has the subset {1, 2, 3}.

## Number of Subsets

A formula to find the total number of subsets for a given set is the product of the set’s cardinality (the number of elements in the set) and the factorial of the number of elements minus 1. For example, the set {1, 2, 3} has a cardinality of 3 and a factorial of 3-1=2, so the total number of subsets for this set is 3×2=6.

## Properties of Power Set

The power set of a set A is the set of all subsets of A.

The power set of a set A is always a subset of the set of all subsets of the power set of A.

The power set of a set A is a set of cardinality 2n, where n is the number of elements in A.

## Cardinality of a Power Set

The cardinality of a power set is the number of possible combinations of elements in a set. For a set with n elements, the cardinality of the power set is 2^n.

## Power Set of Empty Set

The empty set is a set with no elements.

## Power Set of a Countable Set

The power set of a countable set is the set of all subsets of the given set.

## Power Set of an Uncountable Set

There is no set of all possible sets.

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